Unit-IV : Algebraic Structures
DISCRETE MATHEMATICS SUBMITTED BY KOMAL APPLIED SCIENCE Introduction Discrete Mathematics is the part of Mathematics devoted to study of Discrete (Disinct or not connected objects ) Discrete Mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous . As we know Discrete Mathematics is a backbone of mathematics and computer science Scope It Develops our Mathematical Thinking It Improves our problem solving abilities Many Problems can be solved using Discrete mathematics For eg . Sorting the list of Integers Finding the shortest path from home to any destination Drawing a garph within two conditions • We are not allowed to lift your pen. • We are not allowed to repeat edges Algebraic Structures Algebraic systems Examples and general properties Semi groups Monoids Groups Sub groups Algebraic systems N = {1,2,3,4,….. } = Set of all natural numbers. Z = { 0, 1, 2, 3, 4 , ….. } = Set of all integers. Q = Set of all rational numbers. R = Set of all real numbers. Binary Operation: The binary operator * is said to be a binary operation (closed operation) on a non empty set A, if a * b A for all a, b A (Closure property). Ex: The set N is closed with respect to addition and multiplication but not w.r.t subtraction and division. Algebraic System: A set ‘A’ with one or more binary(closed) operations defined on it is called an algebraic system. Ex: (N, + ), (Z, +, – ), (R, +, . , – ) are algebraic systems. Properties Commutative: Let * be a binary operation on a set A. The operation * is said to be commutative in A if a * b= b * a for all a, b in A Associativity: Let * be a binary operation on a set A.
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